However, this picture becomes more subtle as we also considers possible self-interactions of the ULDM particles.
In case that the ULDM particles are subject to attractive self interactions, \(\lambda <0\) , these will lead to instabilities of the dark matter halo and ultimately result in its collapse and the formation of boson stars [1]}.
On the contrary, if the self-interactions are repulsive (\(\lambda >0\) ) these will help to counteract the gravitational collapse of the halo and hence allow for a relaxed mass range for ULDM of \(10^{-22} - 1\)  eV [2]}. These two scenarios are sketched in fig:halos.
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